compar.ou

Ornstein--Uhlenbeck Model for Continuous Characters

This function fits an Ornstein--Uhlenbeck model giving a phylogenetic
tree, and a continuous character. The user specifies the node(s) where
the optimum changes. The parameters are estimated by maximum
likelihood; their standard-errors are computed assuming normality of
these estimates.

Usage

Arguments

a vector giving the number(s) of the node(s) where the
parameter `theta' (the character optimum) is assumed to change. By
default there is no change (same optimum thoughout lineages).

alpha

the value of $\alpha$ to be used when fitting
the model. By default, this parameter is estimated (see details).

Details

The Ornstein--Uhlenbeck (OU) process can be seen as a generalization
of the Brownian motion process. In the latter, characters are assumed
to evolve randomly under a random walk, that is change is equally
likely in any direction. In the OU model, change is more likely
towards the direction of an optimum (denoted $\theta$) with
a strength controlled by a parameter denoted $\alpha$.

The present function fits a model where the optimum parameter
$\theta$, is allowed to vary throughout the tree. This is
specified with the argument node: $\theta$ changes
after each node whose number is given there. Note that the optimum
changes only for the lineages which are descendants of this
node.

Hansen (1997) recommends to not estimate $\alpha$ together
with the other parameters. The present function allows this by giving
a numeric value to the argument alpha. By default, this
parameter is estimated, but this seems to yield very large
standard-errors, thus validating Hansen's recommendation. In practice,
a ``poor man estimation'' of $\alpha$ can be done by
repeating the function call with different values of alpha, and
selecting the one that minimizes the deviance (see Hansen 1997 for an
example).

If x has names, its values are matched to the tip labels of
phy, otherwise its values are taken to be in the same order
than the tip labels of phy.

The user must be careful here since the function requires that both
series of names perfectly match, so this operation may fail if there
is a typing or syntax error. If both series of names do not match, the
values in the x are taken to be in the same order than the tip
labels of phy, and a warning message is issued.

Value

an object of class "compar.ou" which is list with the following
components:

deviancethe deviance (= -2 * loglik).

paraa data frame with the maximum likelihood estimates and
their standard-errors.

callthe function call.

Note

The inversion of the variance-covariance matrix in the likelihood
function appeared as somehow problematic. The present implementation
uses a Cholevski decomposition with the function
chol2inv instead of the usual function
solve.

References

See Also

Aliases

compar.ou

Examples

data(bird.orders)
### This is likely to give you estimates close to 0, 1, and 0
### for alpha, sigma^2, and theta, respectively:
compar.ou(rnorm(23), bird.orders)
### Much better with a fixed alpha:
compar.ou(rnorm(23), bird.orders, alpha = 0.1)
### Let us 'mimick' the effect of different optima
### for the two clades of birds...
x <- c(rnorm(5, 0), rnorm(18, 5))
### ... the model with two optima:
compar.ou(x, bird.orders, node = -2, alpha = .1)
### ... and the model with a single optimum:
compar.ou(x, bird.orders, node = NULL, alpha = .1)
### => Compare both models with the difference in deviances
## with follows a chi^2 with df = 1.